11.1 problem solving using ratios and proportions

11.1 Problem Solving Using Ratios and Proportions

A ratio is the comparison of two numbers written as a fraction.

For example: Your school’s basketball team has won 7 games and lost 3 games. What is the ratio of wins to losses?

Because we are comparing wins to losses the first number in our ratio should be the number of wins and the second number is the number of losses.

The ratio is games won___________games lost

= 7 games_______3 games

= 7__3

In a ratio, if the numerator and denominator are measured in different units then the ratio is called a rate.

A unit rate is a rate per one given unit, like 60 miles per 1 hour.

Example: You can travel 120 miles on 60 gallons of gas. What is your fuel efficiency in miles per gallon?

Rate = 120 miles________60 gallons= ________20 miles

1 gallon

Your fuel efficiency is 20 miles per gallon.

11.1 Problem Solving Using Ratios and Proportions

An equation in which two ratios are equal is called a proportion.

A proportion can be written using colon notation like this

a:b::c:d

or as the more recognizable (and useable) equivalence of two fractions.

a___ ___=b

cd

11.1 Problem Solving Using Ratios and Proportions

a:b::c:d a___ ___=b

cd

When Ratios are written in this order, a and d are the extremes, or outside values, of the proportion, and b and c are the means, or middle values, of the proportion.

Extremes Means

11.1 Problem Solving Using Ratios and Proportions

To solve problems which require the use of a proportion we can use one of two properties.

The reciprocal property of proportions.

If two ratios are equal, then their reciprocals are equal.

The cross product property of proportions.

The product of the extremes equals the product of the means

11.1 Problem Solving Using Ratios and Proportions

x

35

3

5

3535 x

1055 x

11.1 Problem Solving Using Ratios and Proportions

21x

9

62

x

x 629

x618

x3

11.1 Problem Solving Using Ratios and Proportions

Solve:1

21

xx

x – 1 = 2x

x = –1

xx 2)1(1

11.1 Problem Solving Using Ratios and Proportions

Solve:xx

x 1

12

x2 = -2x - 1

x2 +2x + 1= 0

)12(12 xx

11.1 Problem Solving Using Ratios and Proportions

(x + 1)(x + 1)= 0

(x + 1) = 0 or (x + 1)= 0 x = -1

11.2 Problem Solving Using Percents

Percent means per hundred, or parts of 100 When solving percent problems, convert the

percents to decimals before performing the arithmetic operations

11.2 Problem Solving Using Percents

• What is 20% of 50?• x = .20 * 50• x = 10

• 30 is what percent of 80?• 30 = x * 50• x = 30/50 = .6 = 60%

11.2 Problem Solving Using Percents

• 12 is 60% of what?• 12 = .6x• x = 12/.6 = 20

• 40 is what percent of 300?• 40 = x * 300• x = 40/300 = .133… = 13.33%

11.2 Problem Solving Using Percents

What percent of the region is shaded?

60

40

10

10

100 is what percent of 2400?

100 = x * 2400?

x = 100/2400

x = 4.17%

11.3 Direct and Inverse Variation

Direct VariationThe following statements are equivalent:

y varies directly as x. y is directly proportional to x. y = kx for some nonzero constant k.

k is the constant of variation or the constant of proportionality

11.3 Direct and Inverse Variation

Inverse Variation

The following statements are equivalent:

y varies inversely as x. y is inversely proportional to x. y = k/x for some nonzero constant k.

11.3 Direct and Inverse Variation

If y varies directly as x, then y = kx.If y = 10 when x = 2 , then what is the value of y when x = 8?x and y go together. Therefore, by substitution 10 = k(2).What is the value of k? 10 = 2k

10 = 2k

5 = k

11.3 Direct and Inverse Variation

k = 5

Replacing k with 5 gives us y = 5x

What is y when x = 8 ?

y = 5(8)

y = 40

11.3 Direct and Inverse Variation

If y varies inversely as x, then xy = k.

If y = 6 when x = 4 , then what is the value of y when x = 8?

x and y go together. Therefore, by substitution (6)(4) = k.

What is the value of k?

24 = k

11.3 Direct and Inverse Variation

k = 24

Replacing k with 24 gives us xy = 24

What is y when x = 8 ?

8y = 24

y = 3

y = kx

00 5 10 15 20

5

10

15

Direct variation

11.3 Direct and Inverse Variation

y = 2x

••

••

xy= k

00 5 10 15 20

5

10

15 •

••

• •

xy= 16

Inverse Variation

11.3 Direct and Inverse Variation

11.5 Simplifying Rational Expressions

Define a rational expression. Determine the domain of a rational

function. Simplify rational expressions.

Rational numbers are numbers that can be written as fractions.

Rational expressions are algebraic fractions of the form P(x) , where P(x) and Q(x) Q(x) are polynomials and Q(x) does not equal zero.

Example:

3x 2 2x 1

4 x 1

11.5 Simplifying Rational Expressions

P(x) ; Since division by zero is not Q(x) possible, Q(x) cannot equal zero.

The domain of a function is all possible values of x.

For the example , 4x + 1 ≠ 0

so x ≠ -1/4.

14

123 2

x

xx

11.5 Simplifying Rational Expressions

The domain of is

all real numbers except -1/4.

14

123 2

x

xx

Domain = {x|x ≠ -1/4}

11.5 Simplifying Rational Expressions

Find domain of 65

122

xx

x

Domain = {x | x ≠ -1, 6}

Solve: x 2 –5x – 6 =0

(x – 6)(x + 1) = 0The excluded values are x = 6, -1

11.5 Simplifying Rational Expressions

To simplify rational expressions, factor the numerator and denominator completely. Then reduce.

Simplify:

32244

12222

2

xx

xx

11.5 Simplifying Rational Expressions

864

62

32244

12222

2

2

2

xx

xx

xx

xxFactor:

244

232

xx

xxReduce:

2

42

3

x

x

11.5 Simplifying Rational Expressions

Simplify:

x

x

2

2

Factor –1 out of the denominator: x

x

21

2

21

2

x

x

11.5 Simplifying Rational Expressions

Reduce: 21

2

x

x

11

1

11.5 Simplifying Rational Expressions

11.5 Simplifying Rational Expressions

Multiply rational expressions. Divide rational expressions

To multiply, factor each numerator and denominator completely.

Reduce Multiply the numerators and multiply the

denominators. Multiply:

213

12

209

15 2

2

2

x

xx

xx

x

11.5 Simplifying Rational Expressions

xx

xx

xx

x

213

12

209

152

2

2

2

Factor:

73

34

54

15 2

xx

xx

xx

xReduce: 5

7

3

5

5

x

x

x

x

11.5 Simplifying Rational Expressions

7

3

5

5

x

x

x

xMultiply: 75

35

xx

xx

3512

1552

2

xx

xx

11.5 Multiplying and Dividing

To divide, change the problem to multiplication by writing the reciprocal of the divisor. (Change to multiplication and flip the second

fraction.)

Divide:

54

62

1

322

2

2

2

xx

xx

x

xx

11.6 Multiplying and Dividing

62

54

1

322

2

2

2

xx

xx

x

xx

54

62

1

322

2

2

2

xx

xx

x

xx

Change to multiplication:

Factor completely:

232

51

11

132

xx

xx

xx

xx

11.5 Multiplying and Dividing

232

51

11

132

xx

xx

xx

xxReduce:

2

5

x

xMultiply:

11.5 Multiplying and Dividing

11.7 Dividing Polynomials

Dividing a Polynomial by a MonomialLet u, v, and w be real numbers, variables or

algebraic expressions such that w ≠ 0.

w

v

w

u

w

vu

.1

w

v

w

u

w

vu

.2

11.7 Dividing Polynomials

x

xxx

3

9612 23 324 2 xx

c

ccc

9

452718 24 532 3 cc

11.7 Dividing Polynomials

)2()124( 2 xxx Use Long Division

1242 2 xxx

x

x2 -2x6x - 12

+ 6

6x - 12

0

Note: (x + 6) (x – 2) =x2 + 4x - 12

11.7 Dividing Polynomials

)1()24( 2 xxx Use Long Division

141 2 xxx

x

x2 - x5x - 1

+ 5

5x - 5 4

1

4545

xxx orr

11.7 Dividing Polynomials

)2()12( 3 xxx Note: x2 term is missing

1202 23 xxxx

x2

x3 + 2x2

-2x2 + 2x

- 2x

-2x2 – 4x 6x - 1

-

+ 6

6x + 12-

-13

13622 rxx

2

13622

x

xxor

xx

12

2

13

LCD: 2x

Multiply each fraction through by the LCD

x

xx

x

x 12*2

2

23*2

246 x

18 x

18x Check your solution!Check your solution!

18

12

2

1

18

3

1293

11.8 Solving Rational Equations

Solve. 1

54

1

5

xx

xLCD: ?LCD: (x+1)

)1(

)1(5)1(4

)1(

)1(5

x

xx

x

xx

5445 xx145 xx

1x

Check your solution!Check your solution!

11

54

11

)1(5

0

54

0

5

?

No Solution!No Solution!

11.8 Solving Rational Equations

Solve. 14

6

2

232

xx

x

Factor 1st!

1)2)(2(

6

2

23

xxx

x

LCD: (x + 2)(x - 2)

)2)(2()2)(2(

)2)(2(6

)2(

)2)(2)(23(

xx

xx

xx

x

xxx

42264263 22 xxxxxx2443 22 xxx

0642 2 xx0322 xx

0)1)(3( xx01or 03 xx

1or 3 xx

Check your solutions!Check your solutions!

11.8 Solving Rational Equations

Short Cut!

When there is only fraction on each side When there is only fraction on each side of the =, just cross multiply as if you are of the =, just cross multiply as if you are solving a proportion.solving a proportion.

11.8 Solving Rational Equations

Example: Solve.

4

1

4

32

xxx

xx 42 12342 xxx

0122 xx

0)3)(4( xx

03or 04 xx

3or 4 xx

Check your solutions!Check your solutions!

)4(3 x

11.8 Solving Rational Equations

Solve.1

2

22

62

x

x

xx

)1(6 x)2)(1(2 xxx

6)2(2 xx3)2( xx

0322 xx

0)1)(3( xx

01or 03 xx

1or 3 xx

1

2

)1(2

6

x

x

xx

Check your solutions!Check your solutions!

11.8 Solving Rational Equations